dynamics of families of maps tangent to the identitydcheragh/ppad/slides/abate.pdf · dynamics of...

DYNAMICS OF FAMILIES OF MAPS TANGENT TO THE

IDENTITY

Marco Abate

Dipartimento di MatematicaUniversità di Pisa

Parameter problems in analytic dynamics

Imperial College, London, June 27, 2016

MARCO ABATE (UNIVERSITÀ DI PISA) MAPS TANGENT TO THE IDENTITY LONDON 2016 1 / 32

INTRODUCTION

INTRODUCTION

A holomorphic germ f : (Cn,O)→ (Cn,O) is tangent to the identity ifdfO = id, that is if it can be written as

f (z) = z + Pν+1(z) + · · ·

where ν + 1 ≥ 2 is the order of f , and Pν+1 6≡ O is a n-uple of homogeneouspolynomials of degree ν + 1 ≥ 2.


INTRODUCTION

INTRODUCTION

A holomorphic germ f : (Cn,O)→ (Cn,O) is tangent to the identity ifdfO = id, that is if it can be written as

f (z) = z + Pν+1(z) + · · ·

where ν + 1 ≥ 2 is the order of f , and Pν+1 6≡ O is a n-uple of homogeneouspolynomials of degree ν + 1 ≥ 2.

Goal: to describe (at least topologically) the dynamics in a full neighborhoodof the origin.


INTRODUCTION

INTRODUCTION (n = 1)

Leau-Fatou flower theorem (1920).


INTRODUCTION


f (z) = z− z3


INTRODUCTION


Leau-Fatou flower theorem (1920).Remark: the number of (attracting or repelling) petals is equal to ν.


INTRODUCTION



Camacho’s theorem (1978): the germ f is topologically locally conjugated tothe time-1 map f0 of the homogeneous vector field zν+1 ∂

∂z , given by

f0(z) =z

(1− νzν)1/ν .


INTRODUCTION



Camacho’s theorem (1978): the germ f is topologically locally conjugated tothe time-1 map f0 of the homogeneous vector field zν+1 ∂

∂z , given by

f0(z) =z

(1− νzν)1/ν .

Thus in dimension one the topological local dynamics is completelydetermined by the order, and time-1 maps of homogeneous vector fieldsprovide a complete list of models.


INTRODUCTION

INTRODUCTION (n ≥ 2)

Aim of this talk is to advertise a geometric approach that in principle mightlead to a description of the local topological dynamics in a full neighborhoodof the origin for generic germs — and that surely works for time-1 maps of(even non-generic) homogeneous vector fields.


INTRODUCTION


Aim of this talk is to advertise a geometric approach that in principle mightlead to a description of the local topological dynamics in a full neighborhoodof the origin for generic germs.

The ingredients we are going to use are:

a singular holomorphic foliation in Riemann surfaces of Pn−1(C);

two meromorphic connections defined along the leaves of the foliation;

the real geodesic flow along the leaves induced by the connections.


INTRODUCTION



Results already obtained:

description of the dynamics for many families of examples;

discovery of unexpected examples, and explanation of previously knownpuzzling examples;

explanation of why the case n ≥ 3 is substantially more difficult than thecase n = 2;

suggestion of many related (and interesting) open questions.


INTRODUCTION



Results already obtained:

description of the dynamics for many families of examples;

discovery of unexpected examples, and explanation of previously knownpuzzling examples;

explanation of why the case n ≥ 3 is substantially more difficult than thecase n = 2;

suggestion of many related (and interesting) open questions.

Joint work with F. Tovena (Roma Tor Vergata) and F. Bianchi (Pisa-Toulouse).


INTRODUCTION


A parabolic curve for a germ f tangent to the identity is a injectiveholomorphic curve ϕ : Ω→ U \ O such that:

Ω ⊂ C is a simply connected domain with 0 ∈ ∂Ω;

ϕ is continuous at 0 and ϕ(0) = O;

ϕ(Ω) is f -invariant;

f k|ϕ(Ω) converges to O as k→ +∞.


INTRODUCTION


A parabolic curve for a germ f tangent to the identity is a injectiveholomorphic curve ϕ : Ω→ U \ O such that:

Ω ⊂ C is a simply connected domain with 0 ∈ ∂Ω;

ϕ is continuous at 0 and ϕ(0) = O;

ϕ(Ω) is f -invariant;

f k|ϕ(Ω) converges to O as k→ +∞.

Let [·] : Cn \ O → Pn−1(C) be the canonical projection.

A parabolic curve ϕ is tangent to [v] ∈ Pn−1(C) if [ϕ(ζ)]→ [v] as ζ → 0.

A Fatou flower is a set of ν disjoint parabolic curves tangent to the samedirection [v], where ν + 1 is the order of f .


INTRODUCTION


Let f (z) = z + Pν+1(z) + · · · .A direction [v] ∈ Pn−1(C) is characteristic if Pν+1(v) = λv for some λ ∈ C;it is degenerate if λ = 0, non-degenerate otherwise.

THEOREM (ÉCALLE, 1985; HAKIM, 1998)

Let f : (Cn,O)→ (Cn,O) be tangent to the identity at O ∈ Cn, and[v] ∈ Pn−1(C) a non-degenerate characteristic direction. Then f admits aFatou flower tangent to [v].


INTRODUCTION



Remark: f is dicritical if all directions are characteristic.




INTRODUCTION






INTRODUCTION




Parabolic curves are 1-dimensional objects inside an n-dimensional space.


INTRODUCTION





Hakim (1998) has given sufficient conditions for the existence ofk-dimensional parabolic manifolds. Her work has been later extended andgeneralized; see, e.g., Vivas (2012), Rong (2014), Lapan (2015), . . .


INTRODUCTION





Hakim (1998) has given sufficient conditions for the existence ofk-dimensional parabolic manifolds. Her work has been later extended andgeneralized; see, e.g., Vivas (2012), Rong (2014), Lapan (2015), . . .

But even when k = n these techniques are not enough for describing thedynamics in a full neighborhood of the origin; new techniques are needed.


GEOMETRY OF FIXED POINT SETS

BLOWING UP

Let π : (M, S)→ (Cn,O) be the blow-up of the origin in Cn. The exceptionaldivisor S = π−1(O) can be identified with Pn−1(C).

Any germ fo : (Cn,O)→ (Cn,O) tangent to the identity can be lifted to aholomorphic self-map f : (M, S)→ (M, S) fixing pointwise the exceptionaldivisor.

To study the dynamics of fo in a neighborhood of the origin is equivalent tostudy the dynamics of f in a neighborhood of S; e.g., (characteristic)directions for fo becomes (special) points in S.



ORDER OF CONTACT

Let f : M → M be a holomorphic self-map of a complex n-dimensionalmanifold M leaving a complex smooth hypersurface S ⊂ M pointwise fixed(actually, it suffices having f defined in a neighborhood of S).

We denote by OM the sheaf of germs of of holomorphic functions on M, andby IS the ideal subsheaf of germs of holomorphic functions vanishing on S.

REMARK

If fo has order ν + 1 then

νf =

ν if fo is non-dicritical,ν + 1 if fo is dicritical.



ORDER OF CONTACT

Let f : M → M be a holomorphic self-map of a complex n-dimensionalmanifold M leaving a complex smooth hypersurface S ⊂ M pointwise fixed.We denote by OM the sheaf of germs of of holomorphic functions on M, andby IS the ideal subsheaf of germs of holomorphic functions vanishing on S.Given p ∈ S and h ∈ OM,p, set

νf (h; p) = maxµ ∈ N

∣∣ h f − h ∈ IµS,p.

The order of contact of f with S is

νf = minνf (h; p) | h ∈ OM,p .It is independent of p.

REMARK


νf =




ORDER OF CONTACT

Let f : M → M be a holomorphic self-map of a complex n-dimensionalmanifold M leaving a complex smooth hypersurface S ⊂ M pointwise fixed.

The order of contact of f with S is

νf = minνf (h; p) | h ∈ OM,p .

It is independent of p.

REMARK


νf =




CANONICAL MORPHISM

In coordinates (U, z) adapted to S, that is such that S ∩ U = z1 = 0, settingf j = zj f we can write

f j(z) = zj + (z1)νf gj(z) ,

where z1 does not divide at least one gj, for j = 1, . . . , n.



CANONICAL MORPHISM

In coordinates (U, z) adapted to S, that is such that S ∩ U = z1 = 0, settingf j = zj f we can write

f j(z) = zj + (z1)νf gj(z) ,

where z1 does not divide at least one gj, for j = 1, . . . , n.

The gj’s depend on the local coordinates. However, if we set

Xf =n∑

j=1

gj ∂

∂zj ⊗ (dz1)⊗νf

then Xf = Xf |S is independent of the local coordinates, and defines a globalcanonical section of the bundle TM|S ⊗ (N∗S )⊗νf , where NS is the normalbundle of S in M, and thus a canonical morphism Xf : N⊗νf

S → TM|S.



CANONICAL FOLIATION

We say that f is tangential if the image of Xf is contained in TS. In coordinatesadapted to S, this is equivalent to requiring g1|S ≡ 0, that is to z1|g1.

REMARK

When n = 2, S is a Riemann surface; so the canonical foliation reduces to thedata of its singular points. This is the reason why (as we’ll see) the dynamicsin dimension 2 is substantially simpler to study than the dynamics indimension n ≥ 3.



CANONICAL FOLIATION


REMARK

fo is non-dicritical if and only if f is tangential. So the tangential case is themost interesting one.

REMARK




CANONICAL FOLIATION


We say that p ∈ S is singular for f if it is a zero of Xf , and we writep ∈ Sing(f ). We set So = S \

(Sing(S) ∪ Sing(f )

).

REMARK




CANONICAL FOLIATION




).

REMARK

[v] ∈ S = Pn−1(C) is singular for f if and only if it is a characteristic directionof fo.

REMARK




CANONICAL FOLIATION

We say that f is tangential if the image of Xf is contained in TS. In coordinatesadapted to S, this is equivalent to requiring g1|S ≡ 0, that is to z1|g1.We say that p ∈ S is singular for f if it is a zero of Xf , and we writep ∈ Sing(f ). We set So = S \


).

PROPOSITION

If f is tangential and p ∈ So is not singular, then no infinite orbit of f can stayclose to p, that is there is a neighborhood U ⊂ M of p such that for everyz ∈ U there exists k0 > 0 such that f k0(z) /∈ U or f k0(z) ∈ S.

REMARK




CANONICAL FOLIATION




).

Since S is a hypersurface, N⊗νfS has rank one; therefore if f is tangential then

the image of Xf yields a canonical foliation Ff , which is a singularholomorphic foliation of S in Riemann surfaces.

REMARK




PARTIAL MEROMORPHIC CONNECTIONS

Assume we have a complex vector bundle F on a complex manifold S, and amorphism X : F → TS. Let E be another complex vector bundle on S, anddenote by E (respectively, F) the sheaf of germs of holomorphic sections of E(respectively, F).A partial meromorphic connection on E along X is a C-linear map∇ : E → F∗ ⊗ E satisfying the Leibniz condition

∇(hs) = (dh X)⊗ s + h∇s

for every h ∈ OS and s ∈ E . In other words, we can differentiate the sectionsof E only along directions in X(F). The poles of the connection are the pointswhere X is not injective.




In the tangential case, we can take F = N⊗νfS and X = Xf . Then we get:

a partial meromorphic connection∇ on E = NS along Xf by setting

∇u(s) = π([Xf (u), s]|S

)where: s ∈ NS; u ∈ N⊗νf

S ; π : TM,S → NS is the canonical projection; sis any element in TM,S such that π(s) = s; and u is any element of T ⊗νf

M,Ssuch that π(u) = u. Small miracle: ∇ is independent of all the choices.

a partial meromorphic connection, still denoted by∇, on N⊗νfS along Xf ;

a partial meromorphic connection∇o on the tangent bundle to thefoliation Ff along the identity





a partial meromorphic connection∇ on E = NS along Xf


a partial meromorphic connection∇o on the tangent bundle to thefoliation Ff along the identity







a partial meromorphic connection∇o on the tangent bundle to thefoliation Ff along the identity by setting

∇ovs = Xf

(∇X−1

f (v)X−1f (s)

).

Notice that∇o induces a (classical) meromorphic connection on eachleaf of the canonical foliation.







a partial meromorphic connection∇o on the tangent bundle to thefoliation Ff along the identity.

In local coordinates (U, z) adapted to S (that is, U ∩ S = z1 = 0) and to Ff

(that is a leaf is given by z3 = cst., . . . , zn = cst.), ∇ is represented by themeromorphic 1-form

η = − νf1g2∂g1

∂z1

∣∣∣∣S

dz2 ,

while∇o is represented by the meromorphic 1-form

ηo = η − 1g2∂g2

∂z2

∣∣∣∣S

dz2 .



GEODESICS

A geodesic is a smooth curve σ : I → So, with I ⊆ R, such that the image of σis contained in a leaf of Ff and

∇oσ′σ′ ≡ O .

PROPOSITION

σ is a geodesic for∇o if and only if X−1(σ′) is an integral curve of G.



GEODESICS


∇oσ′σ′ ≡ O .

If ηo = k dz2 is the form representing∇o in suitable coordinates then σ is ageodesic if and only if

σ′′ + (k σ)(σ′)2 = 0 .

Notice that k is meromorphic.

PROPOSITION




GEODESICS


∇oσ′σ′ ≡ O .


σ′′ + (k σ)(σ′)2 = 0 .

The geodesic field G on the total space of N⊗νfS is given by

G =n∑

p=2

gp|S v∂

∂zp + νf∂g1

∂z1

∣∣∣∣S

v2 ∂

∂v,

where (z2, . . . , zn; v) are local coordinates on N⊗νfE . It is globally defined!

PROPOSITION




GEODESICS


∇oσ′σ′ ≡ O .


σ′′ + (k σ)(σ′)2 = 0 .

The geodesic field G on the total space of N⊗νfS is given by

G =n∑

p=2

gp|S v∂

∂zp + νf∂g1

∂z1

∣∣∣∣S

v2 ∂

∂v.

PROPOSITION



DYNAMICS

HEURISTIC PRINCIPLE

Heuristic guiding principle: the dynamics of the geodesic flow represents thedynamics of f in a neighborhood of S, at least in generic cases.


DYNAMICS

HEURISTIC PRINCIPLE


When f comes from a fo tangent to the identity, “generic" means “when foonly has non-degenerate characteristic directions."


DYNAMICS

HEURISTIC PRINCIPLE


This becomes a rigorous statement, valid even in non-generic situations, whenf comes from the time-1 map of a homogeneous vector field.


DYNAMICS

HOMOGENEOUS VECTOR FIELDS

A homogeneous vector field of degree ν + 1 ≥ 2 on Cn is given by

Q = Q1 ∂

∂z1 + · · ·+ Qn ∂

∂zn

where Q1, . . . ,Qn are homogeneous polynomials in z1, . . . , zn of degree ν+ 1.We say that Q is non-dicritical if it is not a multiple of the radial vector field.


DYNAMICS



Q = Q1 ∂

∂z1 + · · ·+ Qn ∂

∂zn


The time-1 map of a homogeneous vector field of degree ν + 1 is aholomorphic self-map of Cn tangent to the identity at the origin of orderν + 1, dicritical if and only if Q is dicritical.


DYNAMICS



Q = Q1 ∂

∂z1 + · · ·+ Qn ∂

∂zn


The time-1 map of a homogeneous vector field of degree ν + 1 is aholomorphic self-map of Cn tangent to the identity at the origin of orderν + 1, dicritical if and only if Q is dicritical.

A characteristic leaf is a Q-invariant line Lv = Cv ⊂ Cn. A line Lv is acharacteristic leaf if and only if [v] is a characteristic direction of the time-1map of Q. The dynamics of Q inside a characteristic leaf is 1-dimensional andeasy to study.


DYNAMICS


THEOREM (A.-TOVENA, 2011)Let Q be a homogeneous vector field in Cn of degree ν + 1 ≥ 2. Let S be theexceptional set in the blow-up of the origin in Cn, and denote by π : N⊗νS → Sand by [·] : Cn \ O → Pn−1(C) the canonical projections. Then there existsa ν-to-1 holomorphic covering map χν : Cn \ O → N⊗νS \ S such thatπ χν = [·] and dχν(Q) = G.

Therefore:

(I) γ is a real integral curve of G (outside the characteristic leaves) if andonly if χν γ is an integral curve of G;

(II) if γ is a real integral curve then [γ] is a geodesic;

(III) every geodesic in Pn−1(C) is covered by exactly ν integral curves of Q.


DYNAMICS


THEOREM (A.-TOVENA, 2011)Let Q be a homogeneous vector field in Cn of degree ν + 1 ≥ 2. Let S be theexceptional set in the blow-up of the origin in Cn, and denote by π : N⊗νS → Sand by [·] : Cn \ O → Pn−1(C) the canonical projections. Then there existsa ν-to-1 holomorphic covering map χν : Cn \ O → N⊗νS \ S such thatπ χν = [·] and dχν(Q) = G. Therefore:





DYNAMICS






Thus the study of integral curves of homogeneous vector fields is equivalentto the study of geodesics for partial meromorphic connections on Pn−1(C).


DYNAMICS






The geodesic σ(t) = [γ(t)] gives the complex line containing γ(t); the“speed” X−1

f

(σ′(t)

)gives the position of γ(t) in that line. In particular,

γ(t)→ O if and only if X−1(σ′(t)

)→ O.


DYNAMICS


(At least) two main advantages:

1 use of geometric tools (curvature, Gauss-Bonnet, etc.);2 the variables have been separated (in the coefficients of G).


DYNAMICS


(At least) two main advantages:1 use of geometric tools (curvature, Gauss-Bonnet, etc.);

2 the variables have been separated (in the coefficients of G).


DYNAMICS


(At least) two main advantages:1 use of geometric tools (curvature, Gauss-Bonnet, etc.);2 the variables have been separated (in the coefficients of G).


DYNAMICS



Three main steps:

1 study of the global properties of the canonical foliation (only if n ≥ 3);2 study of the global recurrence properties of the geodesics: it depends on

the residues of (the local meromorphic 1-form representing)∇o.3 study of the local behavior of the geodesics near the poles: it depends on

the residues of (the local meromorphic 1-form representing)∇.


DYNAMICS



Three main steps:

1 study of the global properties of the canonical foliation (only if n ≥ 3);

2 study of the global recurrence properties of the geodesics: it depends onthe residues of (the local meromorphic 1-form representing)∇o.

3 study of the local behavior of the geodesics near the poles: it depends onthe residues of (the local meromorphic 1-form representing)∇.


DYNAMICS



Three main steps:


the residues of (the local meromorphic 1-form representing)∇o.

3 study of the local behavior of the geodesics near the poles: it depends onthe residues of (the local meromorphic 1-form representing)∇.


DYNAMICS



Three main steps:


the residues of (the local meromorphic 1-form representing)∇o.3 study of the local behavior of the geodesics near the poles: it depends on

the residues of (the local meromorphic 1-form representing)∇.


DYNAMICS

A POINCARÉ-BENDIXSON THEOREM

THEOREM (A.-TOVENA, 2011, R = P1(C); A.-BIANCHI, 2016, ANY R)

Let σ : [0,T)→ R \ poles be a maximal geodesic for a meromorphicconnection∇o on a compact Riemann surface R. Then:

1 σ tends to a pole p0 of ∇o; or2 σ is closed or accumulates the support of a closed geodesic; or3 σ accumulates a boundary graph of saddle connections; or4 the ω-limit set of σ has non-empty interior and non-empty boundary

consisting of boundary graphs of saddle connections; or5 σ is dense in R; or6 σ self-intersects infinitely many times.

COROLLARY

If γ is a recurrent integral curve of a homogeneous vector field then γ isperiodic or [γ] intersects itself infinitely many times.


DYNAMICS




1 σ tends to a pole p0 of ∇o; or

2 σ is closed or accumulates the support of a closed geodesic; or3 σ accumulates a boundary graph of saddle connections; or4 the ω-limit set of σ has non-empty interior and non-empty boundary


COROLLARY



DYNAMICS




1 σ tends to a pole p0 of ∇o; or2 σ is closed or accumulates the support of a closed geodesic; or

3 σ accumulates a boundary graph of saddle connections; or4 the ω-limit set of σ has non-empty interior and non-empty boundary


COROLLARY



DYNAMICS




1 σ tends to a pole p0 of ∇o; or2 σ is closed or accumulates the support of a closed geodesic; or3 σ accumulates a boundary graph of saddle connections; or

4 the ω-limit set of σ has non-empty interior and non-empty boundaryconsisting of boundary graphs of saddle connections; or

5 σ is dense in R; or6 σ self-intersects infinitely many times.

COROLLARY



DYNAMICS





consisting of boundary graphs of saddle connections; or

5 σ is dense in R; or6 σ self-intersects infinitely many times.

COROLLARY



DYNAMICS





consisting of boundary graphs of saddle connections; or5 σ is dense in R; or

6 σ self-intersects infinitely many times.

COROLLARY



DYNAMICS






COROLLARY



DYNAMICS






A recurring geodesic is closed, dense or self-intersects infinitely many times.

COROLLARY



DYNAMICS






Closed does not mean periodic.

COROLLARY



DYNAMICS






A saddle connection is a geodesic connecting two poles.

COROLLARY



DYNAMICS






Case (4) cannot happen when R = P1(C). We do not have examples of cases(3) or (4).

COROLLARY



DYNAMICS






We have examples of case (5) when R is a torus, and examples of case (6)when R = P1(C). We do not know whether (6) implies (5). If R = P1(C)then (5) might happen only in case (6).

COROLLARY



DYNAMICS






Case (1) is generic; cases (2), (3), (4) and (6) can happen only if the poles ofthe connection satisfy some necessary conditions expressed in terms of theresidues of∇o.

COROLLARY



DYNAMICS






If R = P1(C), closed geodesics or boundary graphs of saddle connections canappear only if the real part of the sum of some residues is −1; a similarcondition holds for R generic.

COROLLARY



DYNAMICS






If R = P1(C) geodesics self-intersecting infinitely many times can appearonly if the real part of the sum of some residues belongs to(−3/2,−1) ∪ (−1,−1/2); a similar condition holds for R generic.

COROLLARY



DYNAMICS






We have a less precise statement for non-compact Riemann surfaces.

COROLLARY



DYNAMICS






Main tools for the proof:∇o is flat;Gauss-Bonnet theorem relating geodesics and residues;a Poincaré-Bendixson theorem for smooth flows.

COROLLARY



DYNAMICS






COROLLARY



DYNAMICS

LOCAL BEHAVIOR NEAR THE POLES (n = 2)

In dimension 2

G = g2|S v∂

∂z2 + νf∂g1

∂z1

∣∣∣∣S

v2 ∂

∂v.

Three classes of singularities:

apparent if 1 ≤ ordp(g2|S) ≤ ordp

(∂g1

∂z1

∣∣∣S

), that is p is not a pole of∇;

Fuchsian if ordp(g2|S) = ordp

(∂g1

∂z1

∣∣∣S

)+ 1, that is p is a pole of order 1;

irregular if ordp(g2|S) > ordp

(∂g1

∂z1

∣∣∣S

)+ 1, that is p is a pole of order

larger than 1.

THEOREM (A.-TOVENA, 2011)Local holomorphic classification of apparent and Fuchsian singularities, andformal classification of irregular singularities.


DYNAMICS

LOCAL BEHAVIOR NEAR THE POLES: APPARENT

SINGULARITIES (n = 2)

Let p0 ∈ S an apparent singularity, and µ = ordp0(g2|S) ≥ 1. Assume µ = 1(we have a complete statement for µ > 1 too). Take p ∈ So close enoughto p0. Then:

for an open half-plane of initial directions the geodesic issuing from ptends to p0;

for the complementary open half-plane of initial directions the geodesicissuing from p escapes;

for a line of initial directions the geodesic issuing from p is periodic.


DYNAMICS

LOCAL BEHAVIOR NEAR THE POLES: APPARENT


Furthermore, if Q is a homogeneous vector field having a characteristic leafLv such that [v] is an apparent singularity with µ = 1:

no integral curve of Q tends to the origin tangent to [v];there is an open set of initial conditions whose integral curves tend to anon-zero point of Lv;

Q admits periodic integral curves of arbitrarily long periodsaccumulating at the origin.


DYNAMICS

LOCAL BEHAVIOR NEAR THE POLES: FUCHSIAN


Let p0 ∈ S a Fuchsian singularity, and µ = ordp0(g2|S) ≥ 1. Assume µ = 1(we have an almost complete statement for µ > 1 too: resonances appear).Let ρ = Resp0(∇) (necessarily ρ 6= 0 since µ = 1). Take p ∈ So close enoughto p0. Then:

if Re ρ < 0 then p0 is attracting, that is all geodesics σ issuing from pexcept one tends to p0 with X−1

(σ′(t)

)→ O; the only exceptional

geodesic escapes;

if Re ρ > 0 then p0 is repelling, that is all geodesics σ issuing from pexcept one escape, and the only exceptional geodesic tends to p0 in finitetime with

∣∣X−1(σ′(t)

)∣∣→ +∞;

if Re ρ = 0 then issuing from p there are closed geodesics (with “speed”converging either to 0 or to +∞), geodesics accumulating the support ofa closed geodesic, and escaping geodesics.


DYNAMICS

LOCAL BEHAVIOR NEAR THE POLES: FUCHSIAN


Furthermore, if Q is a homogeneous vector field having a characteristic leafLv such that [v] is a Fuchsian singularity with µ = 1 and residue ρ 6= 0:

if Re ρ < 0 there is an open set of initial conditions whose integralcurves tend to the origin tangent to [v];if Re ρ > 0 then no integral curve outside of Lv tends to O tangent to [v];if Re ρ = 0 then there are integral curves converging to O without beingtangent to any direction.


DYNAMICS

LOCAL BEHAVIOR NEAR THE POLES: IRREGULAR


?


DYNAMICS



?Results by Vivas (2012) on the existence of parabolic domains.


DYNAMICS



?Results by Vivas (2012) on the existence of parabolic domains.Possibly Stokes phenomena.


FAMILIES

FAMILIES OF HOMOGENEOUS VECTOR FIELDS (n = 2)

Interesting families of homogenous vector fields of fixed degree ν + 1 can beobtained by fixing the number and (whenever possible) the location of distinctcharacteristic directions, and then using the residues at the characteristicdirections as parameters.


FAMILIES

FAMILIES OF HOMOGENEOUS VECTOR FIELDS (n = 2)

Interesting families of homogenous vector fields of fixed degree ν + 1 can beobtained by fixing the number and (whenever possible) the location of distinctcharacteristic directions, and then using the residues at the characteristicdirections as parameters.Non-dicritical quadratic (ν = 1) homogeneous vector fields can have at most3 distinct characteristic directions. Up to holomorphic conjugation there are:

1 3 distinct quadratic fields with exactly one characteristic direction;2 2 distinct families of quadratic fields with exactly two characteristic

directions, parametrized by the residue at (any) one of them;3 1 family of quadratic fields with three distinct characteristic directions,

parametrized by the residues at (any) two of them.


FAMILIES

TWO DISTINCT CHARACTERISTIC DIRECTIONS

Given ρ ∈ C take

Qρ(z,w) = −ρz2 ∂

∂z+ (1− ρ)zw

∂

∂w.

Two characteristic directions:

[1 : 0]: Fuchsian singularity of order µ = 1 and residue ρ (unless ρ = 0,when it is an apparent singularity of order 1);

[0 : 1]: Fuchsian singularity of order µ = 2 and residue 1− ρ.


FAMILIES

TWO DISTINCT CHARACTERISTIC DIRECTIONS

Qρ(z,w) = −ρz2 ∂

∂z+ (1− ρ)zw

∂

∂w.

If Re ρ < 0 then almost all integral curves converge to the origin tangentto [1 : 0]; each Lv contains exactly one line of exceptional initial valuesof integral curves diverging to infinity tangent to L[0:1].

If Re ρ > 0 the roles of [1 : 0] and [0 : 1] are reversed.

If Re ρ = 0 but ρ 6= 0 then almost all integral curves converge to theorigin without being tangent to any direction; each Lv contains exactlyone line of exceptional initial values of integral curves diverging toinfinity without being tangent to any direction.

If ρ = 0 then almost all integral curves go from one point in L[1:0] toinfinity toward L[0:1]; each Lv contains exactly one real curve ofexceptional initial values of periodic integral curves, and these periodicintegral curves accumulate at the origin.


FAMILIES

THREE DISTINCT CHARACTERISTIC DIRECTIONS

Qρ,τ (z,w) =(−ρz2 + (1− τ)zw

) ∂∂z

+((1− ρ)zw− τw2) ∂

∂w.

Three characteristic directions:

[1 : 0]: Fuchsian singularity of order µ = 1 and residue ρ (unless ρ = 0,when it is an apparent singularity of order 1);

[0 : 1]: Fuchsian singularity of order µ = 1 and residue τ (unless τ = 0,when it is an apparent singularity of order 1);

[1 : 1]: Fuchsian singularity of order µ = 1 and residue 1− ρ− τ (unlessρ+ τ = 1, when it is an apparent singularity of order 1).


FAMILIES

THREE DISTINCT CHARACTERISTIC DIRECTIONS

[1:0]+∞ 1

[1:0]+∞1[1

:0]+∞2

[1:0]

Re(ρ)

Re(τ)

[1:1]

[1:0]+[1:1]

[1:0]+[0:1]

[1:0]+[0:1]+∞1

[1:0

]+[0

:1]+∞1

[0:1]

[0:1]+[1:1]

[0:1]+∞2 [0:1]+[1:1]+∞1

[1:1]+∞1

[0:1

]+∞1

[0:1]+∞ 1

[1:0]+[1:1]+∞ 1

[1:0

]+[1

:1]+∞1

[1:0]+ [0:1]+ ∞2

[0:1]+[1:1]+∞ 1

[1:1

]+∞1

[1:1]+∞ 2

∞3

[1:0]+[1:1] +∞ 2

[0:1]+[1:1] +∞ 2

Re(ρ+τ)=1

1-1/2

-1/2

1

3/2

3/2


PICTURES

MOVIES

Movies!(If there is time. . . )


PICTURES

Q(z, w) = −0.1iz2 ∂∂z + (1 + 0.1i)zw ∂

∂w

-4 -2 2 4

-4

-2

2

4


PICTURES

Q(z, w) = −0.1iz2 ∂∂z + (1 + 0.1i)zw ∂

∂w

-2 -1 1 2

-2

-1

1

2


PICTURES

Q(z, w) = (−0.1z2 + (1− 0.2i)zw) ∂∂z + (1.1zw− 0.2iw2) ∂∂w

-2 -1 1 2

-2

-1

1

2


PICTURES

Q(z, w) = (−13z2 + 2

3zw) ∂∂z + (23zw− 1

3w2) ∂∂w

-4 -2 2 4

-4

-2

2

4


THE END

THANKS!

-4 -2 2 4

-4

-2

2

4


dynamics of families of maps tangent to the identitydcheragh/ppad/slides/abate.pdf · dynamics of...

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